Visualizing Sequences in the Social Sciences

Visualization of Sequences in the Social Sciences

In the social sciences, research questions about categorical sequences are directed both at variation within individual sequences over time and at variation between sequences in a given population. For example, how do family formation processes unfold over time and how similar are they across a population? Variation within individual sequences over time can be measured with different complexity measures (Elzinga 2010). Variation between sequences is a relational property that refers to the similarity of sequences to one another. It can be measured as the dissimilarity between pairs of sequences.

In addition, social scientists are interested in qualitative sequence patterns, that is, the question of “what” the sequence patterns mean substantively, as well as quantitative summary measures that provide answers to the question of “how much” of something is present in the sequences. Qualitative sequence patterns capture sequential processes in the spirit of narrative positivism, which was originally suggested by Abbott (1992) as the conceptual and theoretical foundation of sequential thinking in the social sciences. Quantitative summary measures of sequence information have become increasingly more sophisticated, as the technical development of sequence analysis advanced (e.g., Elzinga 2003, 2010; Gabadinho et al. 2011b). They also more readily bridge across to mainstream regression-based quantitative methodology (Biemann and Wolf 2009).

This interest in the variation within and between sequences as well as in qualitative and quantitative sequence information is challenging for visualization. Graphical tools for complexity reduction used in quantitative time-series analysis, such as line graphs or histograms, are useful to show summary measures of quantitative sequence information, but they cannot visualize qualitative information of sequence patterns (Müller et al. 2008). Sequence states, such as being married or divorced, are categorical and qualitative in nature. It is therefore not possible to calculate averages as meaningful summary measures and plot them over time. One challenge for the visual display of social science sequences is thus to effectively extract the most salient qualitative sequence patterns, while at the same time maintaining quantitative information about the dispersion of sequences around these patterns.

Given this complex and relational data structure, visualization is a potentially powerful tool of exploration and complexity reduction of categorical sequences. Visualization has a long-standing tradition in exploratory and descriptive data analysis (Tukey 1977). As Tukey (1977: v) puts it, “A basic problem about any body of data is to make it more easily and effectively handleable by minds.” To this end, graphs are useful to the extent that they make a simpler description possible and help us look below the described surface and thus make the exploration more effective (Tukey 1977). More recently, visualization has received much attention in social network analysis (Freeman 2002) that examines similarly complex relational data. Network graphs are at the same time a prime example of the pitfalls of visualization (Han 2010). While a good graph will often be the most effective way to convey complex information from data, graphs can also easily be deceptive of actual patterns in the data or distract from main patterns with mere visual “bells and whistles” (Han 2010; Tufte 2001).

According to Tukey (1977: vi), “the greatest value of a picture is when it forces us to notice what we never expected to see.” Tufte (2001) suggested some standards for graphical displays of quantitative information, arguing that the objective of graphs is to communicate complex ideas with clarity, precision, and efficiency. Piccarreta and Lior (2010:166) invoke Tufte’s (2001:51) guidelines when they note that “Graphical excellence is that which gives to us the viewer the greatest number of ideas in the shortest time with the least ink in the smallest space.” Graphical integrity is put forward as the most important principle of graphical excellence. Put simply, graphical integrity is achieved when a graph “tells the truth about the data” (Tufte 2001:105), that is, when visual representation is consistent with the numerical representation in the data. Because visual perception is context dependent, context is essential for graphical integrity. Context dependency is particularly important for group comparisons that place information about one group in the context of information about another group.

Data Representation Graphs: The Sequence Index Plot and Extensions

Table 1 shows descriptive statistics for the example data on family formation, including the percentage of women who ever experienced each focal event and the mean age at the first occurrence of each focal event. Figure 1 shows the sequence index plot and its extensions for the example data.

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Table 1.

Descriptive Information About the Example Data on Family Formation of Women Born 1971 in East and West Germany Between Age 15 and 33 (N = 474).

Sequence State	Percentage Women Ever Experienced the Event	Mean Age at First Occurrence of the Event	Mean Duration in Event (Months)
Single (S)	97.68		62.06
In a relationship (R)	98.31	18.71	53.20
Cohabiting, no child (CNC)	77.00	22.97	29.52
Cohabiting, child (CC)	18.35	25.90	8.69
Married, no child (MNC)	61.81	25.07	17.90
Married, child (MC)	58.86	26.48	43.89
Divorced/widowed (DW)	4.01	27.29	0.75

Source: German Life History Study (GLHS).

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Figure 1.

Data representation graphs: the sequence index plot and extensions.

Data Summarization Graphs: Quantitative Sequence Information

Figure 2 shows summary graphs of quantitative sequence information, such as the average time spent in each categorical state, or the aggregate distribution of states at each time point. They fundamentally differ from the sequence index plot and its extensions, because they do not display individual sequences over time. It is impossible to extract information on individual sequences from them. Instead, they depict summary information for the entire sample of sequences. Essentially, all of these summary plots represent different facets of the same information and can largely be derived from each other. However, they highlight different information about the sequences. A more extensive review of these graphs can be found in Gabadinho et al. (2011a).

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Figure 2.

Data summarization graphs: quantitative sequence information.

State distribution plots (Billari and Piccarreta 2005) aggregate the frequency of each state at each time point. Figure 2(a) shows the state distribution plot for the example data. At the beginning of the sequences only two states occur: being single and in a relationship. At the end of the sequences, all seven states occur. Being in a noncohabiting relationship is the most common state for women in their early 20s. Cohabiting, marriage, and motherhood gain importance as women approach their 30s. State distribution plots give a good overview of the time point–specific distribution of states. However, similar state distribution plots can mask very different individual sequences, because they contain no information about how individuals move back and forth between these states over time.

Kaplan–Meier (KM) survival curves, a standard tool in epidemiology and demography, are useful to visualize the timing of successive stage transitions along a set of sequences. Figure 2(b) shows KM survival curves for transitions into six focal events in the family formation process: (1) into a “relationship” (R), (2) into “cohabitation without a child” (CNC), (3) into “marriage without a child” (MNC), (4) into “cohabitation with a child” (CC), (5) into “marriage with a child” (MC), and (6) into “divorced/widowed” (DW). The KM survival curves S ˆ ( t ) are calculated as (Kaplan and Meier 1958):

S ˆ t = ∏ t i < t n i − d i n i .

Modal state plots (Gabadinho et al. 2011a) are bar graphs of the modal state; that is, the most frequent state at each time point. The height of the bar is proportional to the frequency of the respective state at this time point. Figure 2(c) shows the modal state plot for the example data. Until around age 20, being single is the most frequent state, then being in a relationship, and from age 30 onward being married with children is the most frequent state for women in the example data.

Mean time plots (Gabadinho et al. 2011a) are histograms of the mean time spent in each state across all time points. Similar to the state distribution plot and the modal state plot, the mean time plot in Figure 2(d) shows that German women spent the most time being single and in a noncohabiting relationship between the age of 15 and 34. The least time was spent being divorced. The mean time plot is one of many possibilities to visualize quantitative summary information about sequence structures with standard plots such as bar or line graphs. Other possible summary measures are, for instance, sequence turbulence or complexity measures (Elzinga 2010) as an indicator for variability within individual sequences over time, or the transversal entropy (Billari 2001; Fussel 2005) as an indicator for the variability between sequences at each given time point. For instance, the transversal entropy plot (Billari 2001; Fussel 2005) displays Shannon’s entropy index as a line graph.

In addition, Aassve et al. (2007:381) propose a tree-based sequence representation to graphically represent clusters of sequences, obtained by referring to a given dissimilarity matrix. The central idea is to use medoid sequences to represent clusters. This type of graph is neither an extension of the sequence index plot nor a summary plot of aggregate sequence characteristics. Each cluster is represented by one tree and the medoid sequences in each cluster are the branches of the tree. The nodes in these branches represent the sequence states visited, for instance unemployment or marriage in their application. The length of the arrows that connect two consecutive states is proportional to the average duration of time in one state before experiencing the next (Aassve et al. 2007:280). The tree-based graph capitalizes on graphical representation with single cases to illuminate findings, a mode of representation that is usually reserved for qualitative data analysis. Drawbacks of the tree-based sequence representation are that they are only useful for relatively homogenous groups and that the dispersion of sequences around the medoids is lost in the graphical representation.

Step 1: Sort the Sequences According to a Substantively Meaningful Principle

Initial sorting determines which sequences are grouped together during the partitioning into frequency groups. In the social sciences, sensible sorting principles are the timing of a focal transition of substantive interest, such as the age of first marriage, their scores on an MDS factor, the dissimilarity to the most frequent sequence, or the dissimilarity to a theoretically motivated ideal typical sequence. Choosing the timing of a focal transition of substantive interest can be problematic if sizable proportions of the sample do not experience this transition. For instance, only 60 percent of women were ever married in the observation period covered in our example data. Thus, sorting according to the age of first marriage leaves the sorting of the remaining 40 percent uncontrolled. Choosing the MDS factor, or the dissimilarity to some meaningful sequences as a sorting criterion, circumvents these problems, but warrants careful consideration of the dissimilarity measure chosen. For instance, if one believes the timing of transitions to be particularly meaningful for the substantive process of interest, one may chose Lesnard’s (2010) DHD. If one considers the order in which states occur within the sequence particularly important, one of Elzinga’s (2008) subsequence metrics, or OM with low indel costs and high substitution costs might be a better choice. Finally, the dissimilarity to the most frequent sequence as an additional sorting option is only meaningful, if there is a most frequent sequence that represents a considerable part of the sample.

As a general rule, the sorting principle and dissimilarity measure should be guided by the substantive interest in sequence variation and the empirical distribution of the sequences. The family formation sequences in the example data do not contain a meaningful most frequent sequence because each sequence is unique. Two sensible sorting criteria are the timing of first marriage as a transition of theoretical interest and a score on the first MDS factor extracted from a dissimilarity matrix built using the DHD (Lesnard 2010). The DHD emphasizes the timing of transitions between family formation states, which we consider theoretically important in the process of family formation. The goodness of fit of relative frequency sequence plots using different sorting criteria can be evaluated using the R² and F test proposed in step 6.

Step 2: Divide the Sorted Sample Into K Similarly Sized Frequency Groups

The partitioning into K frequency groups should divide the sample into equal-sized frequency groups. As a rule of thumb, it is advisable to specify around 100 frequency groups if this specification passes the F test proposed below and increase K accordingly otherwise. Up to 100 representative sequences can easily be plotted and interpreted as a conventional sequence index plot without overplotting. Further reducing K is not necessary to avoid overplotting. If the case numbers of the full sample are not evenly dividable by the chosen K, for instance at an N = 474 and K = 100 in our example application, one can easily allow the small difference of one sequence in frequency group sizes to obtain the exact K without eliminating any of the original sequences. In our example application, we can choose 74 frequency groups of five sequences each and 26 frequency groups of four sequences each to cover N = 474 sequences at K = 100.

Note that a medoid can only be chosen for frequency groups that have at least three sequences. Also, the box-and-whisker plot of dissimilarities to the medoid in each frequency group becomes more informative with an increasing number of sequences per frequency group. At the very least each frequency group should have two sequences. If some frequency groups contain only one sequence, it is not clear which sequences to omit for the plot, that is, where on the sorting criterion to place the frequency groups with only one sequence and those with two sequences.

For relatively small samples of a several hundred or fewer sequences, in principle, the sequence heterogeneity will grow with sample size, suggesting a greater number of frequency groups or K with increasing sample size. However, the size of a sample and sample heterogeneity are not necessarily linearly associated, especially for larger samples. When using a good representative national sample (usually of a size of a few thousand or larger), the increase in the amount of heterogeneity with increasing sample sizes should become negligible. Therefore, the strategy of using up to 100 frequency groups with each group representing larger percentages of the sample should still be a feasible solution to extract main sequence patterns effectively. Ultimately, the choice of K should be informed by visual effects and by the R² and F test statistics proposed below.

Step 3: Choose the Medoid Sequence of Each Frequency Group to Represent This Group

In our example application, each frequency group has one unique sequence with a minimum sum of dissimilarities to all other sequences in the respective frequency group with several different distance measures. However, two or more sequences in a frequency group potentially share the same minimum sum of dissimilarities to all other sequences. This is similar to the problem of tied pairwise sequence dissimilarities in cluster analysis using a sequence dissimilarity matrix, that is, when two pairs of sequences have the same dissimilarity to one another (Martin et al. 2008). If such tied minimum sums of dissimilarities occur, one can randomly select one of them, assuming that they represent the frequency group equally well. The choice of the dissimilarity measure should be guided by substantive and theoretical considerations and the R² and F test below.

Step 4: Plot the Selected Representative Sequences as a Sequence Index Plot

This step is straightforward and simply requires plotting the selected medoid sequences for the relative frequency groups as a sequence index plot sorted according to the initial sorting criterion.

Step 5: Plot the Dissimilarity to the Medoid Within Each Frequency Group

The bottom-up approach of sequence frequency plots that represents sequences across the full range of the sample requires the division of the sample into equal-sized frequency groups. Depending on the empirical distribution of the sequences, some of these equal-sized frequency groups represent a very homogenous group, while others represent a more heterogeneous group of sequences. The quality of the representation then is not the same across the whole sample. We therefore complement relative frequency sequence plots with dissimilarity-to-medoid plots.

The dissimilarity-to-medoid plot provides information about the dissimilarity of the medoid sequences to the respective frequency group they represent in a box-and-whisker plot. For instance, if a frequency group contains six sequences, of which one is the medoid, the average dissimilarity is simply calculated as the average dissimilarity among the five remaining sequences to the medoid. The box-and-whiskers present the deviation of the distances to the medoid and give information on the distribution of the sequences in the group. For frequency groups that contain only two sequences, the distance to medoid plot is reduced to showing dots for the distances between the two sequences in each frequency group. Note that the dissimilarity-to-medoid is analogous to criterion B of the representative sequence plot (Figure 1(d)) if the medoid is chosen as the representativeness scale.

Rather than looking at the dissimilarity-to-medoid plot as an absolute quality benchmark, one can think of it more broadly as providing additional information about the data. Since the sequences are ordered according to a meaningful principle in the first place, the dissimilarity-to-medoid plot is informative about sequence dispersion at different locales on this continuum. We will use the example data to demonstrate how this can be useful for the substantive interpretation of sequence variation.

Step 6: Calculate the R² Statistic and F Test to Evaluate the Goodness of Fit

Relative frequency sequence plots require three critical decisions: choice of a sorting criterion, choice of a distance measure to identify the medoids in K frequency groups, and the number of frequency groups K. To evaluate the sensitivity of relative frequency sequence plots to these choices, we propose an R² statistic that can be assessed with an F test based on contrasting distances to the frequency group–specific medoids and the distances to the general medoid. We extend Piccarreta’s (2012) proposition of an R²-type statistic based on the general medoid—the medoid sequence for the entire sample—and calculate an R² that contrasts distances to frequency group medoids to distances to the general medoid.

Let S_i denote the fraction of each relative frequency group of the total group-specific distance squared to the total general distance squared for the same group:

S i = ∑ j = 1 n i q i j 2 ∑ j = 1 n i Q i j 2 ,

1

R 2 = 1 − ∑ i = 1 K S i = 1 − ∑ i = 1 K ∑ j = 1 n i q i j 2 ∑ j = 1 n i Q i j 2 ,

2

F = E S D U S D = R 2 / ( K − 1 ) ( 1 − R 2 ) / ( N − K ) ,

3

Note that the F statistic in equation (3) assumes an F distribution, which is based on the following assumption: The variable involved in equation (3) is an independently distributed chi-square variable. Concerning the assumption, chi-square distributions are sum of squared (deviation/normal) distributions. If the data are dependent, which would be the case for simple pairwise sequence distances, they would not meet the criteria for a chi-square distribution. However, S_i , the basis for calculating the R², is not based on a simple sum of squared distances but instead on a sum of squared distance ratios, that is, the ratio between the distance of a sequence to a local medoid to the distance of this sequence to the global medoid. In principle and on average, a smaller or greater local distance is not related to a smaller or greater global distance. Therefore, the ratio of two kinds of distances randomizes the sum of squared distances. We consider the group-specific summed squared ratios as “cases.” They are a sum of independent squared ratios and thus meet the assumption of the F distribution, that is, we can reasonably assume that they are based on an independent chi-square distribution.

Table 2 shows the R² and F test in equation (3) for different distance measures and sorting criteria for K = 100. In terms of distance measures, we use the DHD, OM with constant substitution costs of 2 and indel costs of half the max subcosts of 1 (see MacIndoe and Abbott 2004), as well as the Longest Common Prefix (LCP) distance, one of Elzinga’s subsequence metrics (Elzinga 2008). Two different sorting criteria are tested: the age of first marriage and the score on the first factor derived by applying MDS with the respective distance measure. Note that one could also use a different distance measure in the MDS and to calculate S_i , if one had a theoretical or substantive justification for doing so.

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Table 2.

Overview of R² and F Test for Different Combinations of Sorting Principle, Distance Measure, and K = 100 Frequency Groups, For Example Data on Family Formation With N = 474.

Sorting Principle	Distance Measure	R 2	F	α Probability
Age of first marriage	DHD	0.423	2.767	<.001
Age of first marriage	OM	0.340	1.949	<.001
Age of first marriage	LCP	0.165	0.747	>.950
MDS score	DHD	0.477	3.449	<.001
MDS score	OM	0.552	4.647	<.001
MDS score	LCP	0.369	2.209	<.001

Note: DHD = dynamic Hamming distance, OM = optimal matching with substitution costs = 2 and indel costs = 1, LCP = longest common prefix distance; MDS = multidimensional scaling.

The R² and F test indicate that sorting on the score of the first MDS factor always fares better than sorting on the age of first marriage (Table 2). Also, the DHD yields the highest R² when the sequences are sorted according to the age of first marriage, whereas the R² is highest using OM when the sequences are sorted according to the score of the first MDS factor. Both the R² values obtained with the DHD and OM improve considerably compared to those obtained with the LCP distance. Family formation is a fairly unidirectional process in which timing is highly consequential. The DHD and OM both give a more sophisticated account of timing than the LCP distance, which focuses more on the order of states. It is also intuitive that the sorting on the factor derived by applying MDS yields better results. It gives a sorting guideline across the full range of the sample, whereas the age of first marriage only provides a sorting guideline for the roughly 60 percent of women who are ever married (see Table 1) in our example data, leaving the remaining sorting uncontrolled.

Based on the R² and F tests in Table 2, sorting according to the score of the first factor derived with MDS and using OM is the best specification of the relative frequency sequence plot. It clearly avoids overplotting with 100 sequences at an R² of .552 and passes the F test at α = .001. This model has enough “explanatory” power in visually summarizing the sequences in the sample. In contrast, the worst specification is given when the sequences are sorted by the age of first marriage with the LCP distance at an R² of .165 that does not pass the F test (Table 2).

Figure 3 illustrates these two best and worst specifications of the relative frequency sequence plot (Table 2). Graphs 3(a) to (c) in the upper panel show the sequence index plot, relative frequency sequence plot, and dissimilarity-to-medoid plot sorted by the score on the first factor derived by applying MDS and using OM. The relative frequency sequence plot provides a condensed and crystallized visualization of the same substantive pattern visible in the original sequence index plot. Thus, it performs well as a complexity reduction to extract the main sequence patterns without notable information loss when combined with the dissimilarity-to-medoid plot. Less ink is used than in the full sequence index plot, thus providing a better data-ink ratio. More importantly, overplotting is avoided.

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Figure 3.

Sequence index plot, worst and best specification of relative frequency (RF) sequence plot and dissimilarity-to-medoid plot of women’s family formation based on R² and F test in Table 2 with K = 100.

The dissimilarity-to-medoid box-and-whisker plot in Figure 3(c) underlines a polarization between women who marry and have children early in the lower part of Figure 3(b) and women who remain single and in noncohabiting relationships until the age of 33 at the top. Family formation sequences are particularly similar to their frequency group medoids with a low mean distance and lower dispersion around this mean at both ends of the continuum from more-to-less traditional family formation. Sequence dispersion is largest in the middle range of the scores on the MDS factor, where family formation sequences change states more frequently including several cohabiting episodes. This supports the polarization hypothesis of women’s family formation, particularly in West Germany, into two homogenous patterns: a traditional early marriage and motherhood pattern and a delayed family formation pattern of women who forgo and delay family formation to establish employment careers (see Buchmann and Kriesi 2011). Together Figure 3(a) to (c) suggests that these patterns are internally relatively homogenous opposites on a continuum derived with MDS.

Quite the contrary, the specification sorted by the age of first marriage and using the LCP dissimilarity fails to convey this information on a polarization of women into two homogeneous patterns located on a continuum from more-to-less traditional family formation. The crucial point is that the visual summary of the set of sequences is distorted. Notably, in the lower panel with plots 3(d) to (f) the state “in a relationship” (R), represented by orange, is not properly represented visually. The medoids mostly summarize and describe the other relationship states.

We can conclude that when correctly specified as indicated by the R² and F test statistics and combined with the dissimilarity-to-medoid plot, the graphical representation of relative frequency sequence plots is consistent with the numerical representation in the data. The context dependency of visual impression based on the order of sequences is made explicit in step 1 of generating relative frequency sequence plots, the choice of a sorting criterion. As noted by Piccarreta (2012:368), there are theoretical justifications for each of the available distance measures. The choice of a distance measure is a substantive and theoretical concern that should be guided by the research question/research questions at hand. It has to be chosen carefully and most applications will benefit from comparing several. The R² and the F test provide a statistical guideline for this comparison. In our example application, substantive and statistical criteria closely correspond in unanimously supporting a specification sorted by the score on the first factor derived by applying MDS and using OM both for the MDS and for the medoid extraction. For this specification, 100 sequences prove sufficient to represent the heterogeneity of the sequences across the full range of the sample while at the same time maintaining a visual sense of the relative frequency of specific types of sequences from more-to-less traditional family formation. Further reducing the number of relative frequency groups K is not necessary, since 100 sequences are not at risk of overplotting and give a good visual representation of the example data on women’s family formation.

Relative frequency sequence plots can also address the context dependency of visual perception in the comparison of groups of unequal sizes by way of relative frequencies. Figure 4 shows sequence index plots, relative frequency sequence plots, and dissimilarity-to-medoid plots separately for East (N = 132) and West German (N = 342) women. We specify 66 frequency groups for both German subsocieties. For West Germany, this amounts to five or six sequences per frequency group at R² = .543 and F = 5.039 that is significant at the 10 percent level (upper tail cutoff = 1.304) using OM and the MDS sorting criterion. The East German sample is divided into 66 frequency groups of two sequences each at R² = .582 and F = 1.416 that is just significant at the 10 percent level (upper tail cutoff = 1.377) using the same specification. Correspondingly, the dissimilarity plot in Figure 4(f) is reduced to a simple dot plot that shows the distance between the two sequences in each frequency group. Allowing more frequency groups for East Germany would entail at least some frequency groups of only one sequence, which are problematic for reasons outlined above. To avoid overplotting, it is not necessary to go below 66 frequency groups. The relative frequency sequence plots with K = 66 pass the F test for both German subsocieties and present a good visual comparison of the salient family formation patterns.

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Figure 4.

Sequence index plot, relative frequency (RF) sequence plot and dissimilarity-to-medoid plot of family formation in East and West Germany, optimal matching, sorted by scores on a factor derived with multidimensional scaling (MDS), K = 66.

The comparison shows that among East German women motherhood in cohabiting relationships is far more common and the sequence of relationship states varies more in terms of timing and order. In West Germany, women polarize into either a traditional or a delayed family formation group. Also, there is much stronger sequencing into a traditional pattern of entering a relationship followed by cohabitation, then marriage, and then motherhood in West Germany compared to East Germany.

In sum, the relative frequency sequence plot is a simple technique for reducing overplotting that maintains the possibility to trace representative individual sequences over time and visualizes qualitative sequence patterns. Combined with the dissimilarity-to-medoid plot, they also provide an explicit visualization of the quantitative information on sequence dispersion. The R² and F test statistics offer guidelines for the joint choice of the sorting criteria, distance measure, and number of frequency groups K. These choices however should also be backed up by theoretical and substantive considerations. As our example above showed, these likely will lead to similar conclusions in most cases. We argue that these plots are particularly useful for analyzing categorical sequences in the study of life courses and careers, because patterning in these types of sequences is often strong but fuzzy—there is strong patterning either into distinct groups or along a continuum as derived by applying MDS but there are few identical sequences.